A Quantum Analogue of the Covering Lemma: Limits of Exact Anonymous Computation

Can quantum communication determine the size of an anonymous network when no size bound is known? Classical impossibility proofs rely on the covering lemma: an execution can be reproduced on a larger network with the same local structure. Quantum interference invalidates this argument - a measurement record possible on a network can become impossible on a cover. We show that the impossibility of exact counting nevertheless survives. Our main result is a quantum analogue of the covering lemma that preserves local possibilities rather than complete executions. Within a cyclic covering family, every local measurement outcome possible at a node remains possible at a corresponding node in every sufficiently large network. This holds even after information has travelled around the smaller network, beyond the reach of a simple locality argument. We extend the principle to the universal covering tree and finite covers with sufficiently long cycles. The proof combines tensor-network methods with graph coverings: a common representation across network sizes turns normalization into a constraint on which local outcomes can occur. For uniform synchronous protocols with independently prepared nodes, this support-lifting principle rules out zero-error counting and even the production of a guaranteed finite size bound on anonymous rings and the other cyclic covering families considered here. It also rules out exact leader election with termination in finite time with probability one on these families. Measurements, classical control, and local memory growing with time are allowed; no predetermined running-time bound is assumed. Quantum interference thus defeats execution lifting without enabling infallible certification of network size.